Given a strip of paper, how might you divide it into fourths without using a ruler? Undoubtedly, you'd fold the strip in half and then in half again to locate the quarter marks. Now suppose that your goal is to divide a strip into sixths. You might start by folding the strip into thirds and …

Continue Reading ›› On the NCTM discussion site myNCTM, there's currently an extended discussion about "Division and multiplication of fractions." As the discussion has continued, I've grown concerned with what I see as a fundamental problem with the way we often introduce multiplication as repeated addition: "Multiplying 4 by 5 means we're combining five groups of four items. …

Continue Reading ›› In my

very first Sine of the Times blog post from January 2012, I wrote about the paucity of fractions that young learners typically encounter in their math classes. While they might construct visual representations of 1/2, 2/3, and 8/12, it's unlikely they'll create models of 7/31, 36/19, or 5/101. That's a shame because without …

Continue Reading ›› In my recent posts, I've introduced interactive models for

comparing fractions and

multiplying fractions. To continue the fraction theme, below (and

here) is a

Web Sketchpad model in which the need for equivalent fractions arises naturally through the rules of a game. The model displays two arrays. Dragging the four points changes the arrays' dimensions. The goal is to drag …

Continue Reading ›› In my previous post, I presented an interactive

Web Sketchpad model for visualizing and solving fraction multiplication problems. This week, I'd like to back up a step and focus on the more fundamental skill of visualizing and reasoning about the size of fractions. The fraction game below (and

here) presents two random fractions at a time …

Continue Reading ›› ## The Web Sketchpad and Math Education Blog